1. Field of the Invention
This invention relates to an interpolating operation method and apparatus for an image signal.
2. Description of the Prior Art
Techniques for photoelectrically reading out an image, which has been recorded on a photographic film, in order to obtain an image signal, carrying out appropriate image processing on the image signal, and then reproducing a visible image by use of the processed image signal, have heretofore been known in various fields.
Also, it has been proposed to use stimulable phosphors in radiation image recording and reproducing systems. Specifically, a radiation image of an object, such as a human body, is recorded on a sheet provided with a layer of the stimulable phosphor (hereinafter referred to as a stimulable phosphor sheet). The stimulable phosphor sheet, on which the radiation image has been stored, is then exposed to stimulating rays, such as a laser beam, which cause it to emit light in proportion to the amount of energy stored thereon during its exposure to the radiation. The light emitted by the stimulable phosphor sheet, upon stimulation thereof, is photoelectrically detected and converted into an electric image signal. The image signal is then processed and used for the reproduction of the radiation image of the object as a visible image on a recording material, such as photographic material, or on a display device, such as a cathode ray tube (CRT) display device. Radiation image recording and reproducing systems, which use stimulable phosphor sheets, are advantageous over conventional radiography using silver halide photographic materials, in that images can be recorded even when the energy intensity of the radiation, to which the stimulable phosphor sheet is exposed, varies over a wide range.
In image recording and reproducing systems, in which an image signal is obtained in the manner described above and a visible image is reproduced from the image signal, in cases where the region of interest in the visible image is to be viewed in more detail, the region of interest is often enlarged and reproduced. In such cases, if the enlargement of the image size is carried out such that the number of the image signal components of the image signal representing the enlarged image may be identical with the number of the image signal components of the original image signal representing the original image, the sharpness of the enlarged image will be recognized to be lower than the sharpness of the original image due to the visual characteristics of persons. Therefore, if the image is merely enlarged and reproduced, an enlarged image having a high sharpness cannot be obtained, and the details of the image cannot be viewed accurately.
In order for the aforesaid problems to be eliminated, a predetermined interpolating operation may be carried out on the original image signal, which has been obtained by reading out an original image, and an interpolation image signal, which is a secondary image signal and is made up of a number of image signal components different from that of the original image signal, may thereby be formed. Specifically, in cases where an enlarged image is to be reproduced, an interpolation image signal, which is made up of a number of image signal components larger than that of the original image signal, may be formed from the interpolating operation. A visible image may then be reproduced from the interpolation image signal. In this manner, the sharpness of the enlarged image can be prevented from becoming low.
As the interpolating operation methods for carrying out interpolating operations on image signals, various methods have heretofore been proposed. Among such methods, the method using third-order spline interpolating functions is popular. With the interpolating operation method using the third-order spline interpolating functions, digital original image signal components (Y.sub.k) in each section are connected by a third-order function {f.sub.k }, and the value of f.sub.k corresponding to a position, at which an interpolation point is set, (i.e., a setting position in each section) is taken as the value of the interpolated image signal component.
The interpolating operations, which pass through the original image signal in the manner described above, can yield an image having a comparatively high sharpness. As such interpolating operations, cubic spline interpolating operations, and the like, are known. How the cubic spline interpolating operations are carried out will be described hereinbelow.
FIG. 4 is an explanatory graph showing how interpolated image signal components are obtained with a conventional cubic spline interpolating operation from original image signal components, which are sampled with a period of an equal interval and represent sampling points (picture elements) arrayed in one direction. As illustrated in FIG. 4, the image signal components (the original image signal components), which have been detected as digital signal components from an original image and represent a series of picture elements X.sub.k-2, X.sub.k-1, X.sub.k, X.sub.k+1, X.sub.k+2, . . . , are respectively represented by Y.sub.k-2, Y.sub.k-1, Y.sub.k, Y.sub.k+1, Y.sub.k+2, . . . A third-order spline interpolating function is set for each of sections X.sub.k-2 .about.X.sub.k-1, X.sub.k-1 .about.X.sub.k, X.sub.k .about.X.sub.k+1, and X.sub.k+1 .about.X.sub.k+2. The spline interpolating functions corresponding to the respective sections are represented by f.sub.k-2, f.sub.k-1, f.sub.k, f.sub.k+1, and f.sub.k+2. The interpolating functions are the third-order functions, in which the position in each section serves as a variable.
Firstly, how the interpolating operation is carried out when a point taken for interpolation (hereinbelow referred to as the interpolation point) X.sub.p falls within the section X.sub.k .about.X.sub.k+1 will be described hereinbelow. The spline interpolating function f.sub.k corresponding to the section X.sub.k .about.X.sub.k+1 is represented by Formula (29). EQU f.sub.k (x)=A.sub.k x.sup.3 +B.sub.k x.sup.2 +C.sub.k x+D.sub.k( 29)
In the cubic spline interpolating operation, it is necessary that the spline interpolating function f.sub.k passes through the original sampling points (picture elements), and that the first-order differential coefficient of the spline interpolating function f.sub.k is continuous between adjacent sections. Therefore, it is necessary for Formulas (7), (8), (9), and (10) to be satisfied. EQU f.sub.k (X.sub.k)=Y.sub.k ( 7) EQU f.sub.k (X.sub.k+1)=Y.sub.k+1 ( 8) EQU f.sub.k '(X.sub.k)=f.sub.k-1 '(X.sub.k) (9) EQU f.sub.k '(X.sub.k+1)=f.sub.k+1 '(X.sub.k+1) (10)
In these formulas, f.sub.k ' represents the first-order differentiation (3A.sub.k x.sup.2 +2B.sub.k x+C.sub.k) of the function f.sub.k.
In the strict sense, the cubic spline interpolating operation contains the continuity conditions of the second-order differential coefficient. However, with continuity conditions of the second-order differential coefficient, the operation formulas become complicated. Therefore, the cubic spline interpolating operation is popularly carried out in the form simplified in the manner described above.
Also, in the cubic spline interpolating operation, it is necessary for the first-order differential coefficient at the picture element X.sub.k to satisfy the condition with respect to the picture elements X.sub.k-1 and X.sub.k+1, which are located before and after the picture element X.sub.k, in that the first-order differential coefficient at the picture element X.sub.k should coincide with the gradient (Y.sub.k+1 -Y.sub.k-1)/(X.sub.k+1 -X.sub.k-1) of the image signal components Y.sub.k-1 and Y.sub.k+1 representing the picture elements X.sub.k-1 and X.sub.k+1. Therefore, it is necessary for Formula (19) to be satisfied. EQU f.sub.k '(X.sub.k)=(Y.sub.k+1 -Y.sub.k-1)/(X.sub.k+1 -X.sub.k-1)(19)
Also, it is necessary for the first-order differential coefficient at the picture element X.sub.k+1 to satisfy the condition with respect to the picture elements X.sub.k and X.sub.k+2, which are located before and after the picture element X.sub.k+1, in that the first-order differential coefficient at the picture element X.sub.k+1 should coincide with the gradient (Y.sub.k+2 -Y.sub.k)/(X.sub.k+2 -X.sub.k) of the image signal components Y.sub.k and Y.sub.k+2 representing the picture elements X.sub.k and X.sub.k+2. Therefore, it is necessary for Formula (20) to be satisfied . EQU f.sub.k '(X.sub.k+1)=(Y.sub.k+2 -Y.sub.k)/(X.sub.k+2 -X.sub.k)(20)
It is herein assumed that the interval (i.e., the lattice interval) of each of sections X.sub.k-2 .about.X.sub.k-1, X.sub.k-1 .about.X.sub.k, X.sub.k .about.X.sub.k+1, and X.sub.k+1 .about.X.sub.k+2 is equal to 1, and the position of the interpolation point X.sub.p, which is taken from the picture element X.sub.k toward the picture element X.sub.k+1, is represented by t (0.ltoreq.t.ltoreq.1). In such cases, from Formulas (29), (7), (8), (9), (10), (19), and (29), the formulas shown below obtain. EQU f.sub.k (0)=D.sub.k =Y.sub.k EQU f.sub.k (1)=A.sub.k +B.sub.k +C.sub.k +D.sub.k =Y.sub.k+1 EQU f.sub.k '(0)=c.sub.k =(Y.sub.k+1 -Y.sub.k--1)/2 EQU f.sub.k '(1)=3A.sub.k +2B.sub.k +C.sub.k =(Y.sub.k+2 -Y.sub.k)/2
Therefore, the formulas shown below obtain. EQU A.sub.k =(Y.sub.k+2 -3Y.sub.k+1 +3Y.sub.k -Y.sub.k-1)/2 EQU B.sub.k =(-Y.sub.k+2 +4Y.sub.k+1 -5Y.sub.k +2Y.sub.k-1)/2 EQU C.sub.k =(Y.sub.k+1 -Y.sub.k-1)/2 EQU D.sub.k =Y.sub.k
As described above, the variable conversion of X=t is carried out, and therefore the spline interpolating function f.sub.k (x) is represented by the formula shown below. EQU f.sub.k (x)=f.sub.k (t)
Therefore, an interpolated image signal component Y.sub.p corresponding to the interpolation point X.sub.p may be represented by Formula (30). EQU Y.sub.p =f.sub.k (t)=A.sub.k t.sup.3 +B.sub.k t.sup.2 +C.sub.k t+D.sub.k( 30)
Substituting the coefficients A.sub.k, B.sub.k, C.sub.k, and D.sub.k into Formula (30) yields ##EQU1## Arranging this formula with respect to the image signal components Y.sub.k-1, Y.sub.k, Y.sub.k+1, and Y.sub.k+2 yields Formula (31). ##EQU2##
The coefficients for the image signal components Y.sub.k-1, Y.sub.k, Y.sub.k+1, and Y.sub.k+2 are referred to as the interpolation coefficients c.sub.k-1, c.sub.k, c.sub.k+1, and c.sub.k+2. Specifically, the interpolation coefficients c.sub.k-1, c.sub.k, c.sub.k+1, and c.sub.k+2, which respectively correspond to the image signal components Y.sub.k-1, Y.sub.k, Y.sub.k+1, and Y.sub.k+2 in Formula (31), may be represented by the formulas shown below. EQU c.sub.k-1 =(-t.sup.3 +2t.sup.2 -t)/2 EQU c.sub.k =(.sup.3 t.sup.3 -5t.sup.2 +2)/2 EQU c.sub.k+1 =(-3t.sup.3 +4t.sup.2 +t)/2 EQU c.sub.k+2 =(t.sup.3 -t.sup.2)/2
The operations described above are repeated for the sections X.sub.k-2 .about.X.sub.k-1, X.sub.k-1 .about.X.sub.k, X.sub.k .about.X.sub.k+1, and X.sub.k+1 .about.X.sub.k+2. In this manner, an interpolation image signal can be obtained, which is made up of image signal components occurring at intervals different from those of the image signal components of the entire original image signal.
As described above, in the cubic spline interpolating operation, it is necessary that the spline interpolating function passes through the original sampling points (picture elements), and that the first-order differential coefficient of the spline interpolating function is continuous between adjacent sections. With the interpolating function for the cubic spline interpolating operation, the interpolation image signal for use in the reproduction of a secondary image (i.e., the image obtained from the interpolating operation), which has a comparatively high sharpness, is obtained. On the other hand, as for a portion in the original image, at which the change in density is gentle, the interpolating operation should preferably be carried out such that a secondary image, in which the sharpness is comparatively low and which is smooth, may be obtained. As the interpolating function for obtaining the interpolation image signal representing the secondary image, in which the sharpness is comparatively low and which is smooth, for example, a B spline interpolating operation function is known. In the B spline interpolating operation, the spline interpolating function need not pass through the original sampling points (picture elements), and it is necessary that the first-order differential coefficient and the second-order differential coefficient {represented by f"(X)} of the spline interpolating function are continuous between adjacent sections.
Specifically, in Formula (29), EQU f.sub.k (x)=A.sub.k x.sup.3 +B.sub.k x.sup.2 +C.sub.k x+D.sub.k( 29)
the conditions shown below should be satisfied. EQU f.sub.k '(X.sub.k)=f.sub.k-1 '(X.sub.k) (9) EQU f.sub.k '(X.sub.k+1)=f.sub.k+1 '(X.sub.k+1) (10) EQU f.sub.k "(X.sub.k)=f.sub.k-1 "(X.sub.k) (32) EQU f.sub.k "(X.sub.k+1)=f.sub.k+1 "(X.sub.k+1) (33)
Also, it is necessary for the first-order differential coefficient at the picture element X.sub.k to satisfy the condition with respect to the picture elements X.sub.k-1 and X.sub.k+1, which are located before and after the picture element X.sub.k, in that the first-order differential coefficient at the picture element X.sub.k should coincide with the gradient (Y.sub.k+1 -Y.sub.k-1)/(X.sub.k+1 -X.sub.k-1) of the image signal components Y.sub.k-1 and Y.sub.k+1 representing the picture elements X.sub.k-1 and X.sub.k+1. Therefore, it is necessary for Formula (19) to be satisfied. EQU f.sub.k '(X.sub.k)=(Y.sub.k+1 -Y.sub.k-1)/(X.sub.k+1 -X.sub.k-1)(19)
Further, it is necessary for the first-order differential coefficient at the picture element X.sub.k+1 to satisfy the condition with respect to the picture elements X.sub.k and X.sub.k+2, which are located before and after the picture element X.sub.k+1, in that the first-order differential coefficient at the picture element X.sub.k+1 should coincide with the gradient (Y.sub.k+2 -Y.sub.k)/(X.sub.k+2 -X.sub.k) of the image signal components Y.sub.k and Y.sub.k+2 representing the picture elements X.sub.k and X.sub.k+2. Therefore, it is necessary for Formula (20) to be satisfied . EQU f.sub.k '(X.sub.k+1)=(Y.sub.k+2 -Y.sub.k)/(X.sub.k+2 -X.sub.k)(20)
In general, the function f(X) may be approximately represented by Formula (34). EQU f(X)=f(0)+f'(0)X+{f"(0)/2}X.sup.2 ( 34)
It is herein assumed that the interval (i.e., the lattice interval) of each of sections X.sub.k-2 .about.X.sub.k-1, X.sub.k-1 .about.X.sub.k, X.sub.k .about.X.sub.k+1, and X.sub.k+1 .about.X.sub.k+2 is equal to 1, and the position of the interpolation point X.sub.p, which is taken from the picture element X.sub.k toward the picture element X.sub.k+1, is represented by t (0.ltoreq.t.ltoreq.1). In such cases, from Formulas (29), (9), (10), (19), (20) (30), (32), (33), and (34), the formulas shown below obtain. EQU f.sub.k '(0)=C.sub.k =(Y.sub.k+1 -Y.sub.k-1)/2 EQU f.sub.k '(1)=3A.sub.k +2B.sub.k +C.sub.k =(Y.sub.k+2 -Y.sub.k)/2 EQU f.sub.k "(0)=Y.sub.k+1 -2Y.sub.k +Y.sub.k-1 =2B
Therefore, the formulas shown below obtain. EQU A.sub.k =(Y.sub.k+2 -3Y.sub.k+1 +3Y.sub.k -Y.sub.k-1)/6 EQU B.sub.k =(Y.sub.k+1 -2Y.sub.k +Y.sub.k-1)/2 EQU C.sub.k =(Y.sub.k+1 -Y.sub.k-1)/2
Since D.sub.k is unknown, it is represented by the formula EQU D.sub.k=( D.sub.1 Y.sub.k+2 +D.sub.2 Y.sub.k+1 +D.sub.3 Y.sub.k +D.sub.4 Y.sub.k-1)/6
As described above, the variable conversion of X=t is carried out, and therefore the spline interpolating function f.sub.k (X) is represented by the formula shown below. EQU f.sub.k (x)=f.sub.k (t)
Therefore, ##EQU3## Arranging this formula with respect to the image signal components Y.sub.k-1, Y.sub.k, Y.sub.k+1, and Y.sub.k+2 yields Formula (35). ##EQU4##
If t is set to be t=1, the formula shown below will obtain. EQU f.sub.k (1)={(D.sub.4 -1)/6}Y.sub.k-1 +{(D.sub.3 -3)/6}Y.sub.k +{(D.sub.2 +3)/6}Y.sub.k+1 +{(D.sub.1 +1)/6}Y.sub.k+2
As for the section X.sub.k+1 .about.X.sub.k+2, Formula (35) may be rewritten as Formula (36) ##EQU5##
If t is set to be t=0, the formula shown below will obtain. EQU f.sub.k+1 (0)=(D.sub.4 /6)Y.sub.k +(D.sub.3 /6)Y.sub.k+1 +(D.sub.2 /6)Y.sub.k+2 +(D.sub.1 /6)Y.sub.k+3
From the continuity condition {f.sub.k (1)=f.sub.k+1 (0)} and the condition in that the coefficients corresponding to the respective original image signal components are equal to each other, D.sub.4 -1=0, D.sub.3 -3=D.sub.4, D.sub.2 +3=D.sub.3, D.sub.1 +1=D.sub.2, and D.sub.1 =0. Therefore, EQU D.sub.k =(Y.sub.k+1 +4Y.sub.k +Y.sub.k-1)/6
Accordingly, Formula (37) obtains. ##EQU6##
Therefore, the interpolation coefficients b.sub.k-1, b.sub.k, b.sub.k+1, and b.sub.k+2, which respectively correspond to the image signal components Y.sub.k-1, Y.sub.k, Y.sub.k+1, and Y.sub.k+2, may be represented by the formulas shown below. EQU b.sub.k-1 =(-t.sup.3 +3t.sup.2 -3t+1)/6 EQU b.sub.k =(3t.sup.3 -6t.sup.2 +4)/6 EQU b.sub.k+1 =(-3t.sup.3 +3t.sup.2 +3t+1)/6 EQU b.sub.k+2 =t.sup.3 /6
The operations described above are repeated for the sections X.sub.k-2 .about.X.sub.k-1, X.sub.k-1 .about.X.sub.k, X.sub.k .about.X.sub.k+1, and X.sub.k+1 .about.X.sub.k+2. In this manner, an interpolation image signal can be obtained, which is made up of image signal components occurring at intervals different from those of the image signal components of the entire original image signal.
In this manner, in cases where a secondary image (an interpolation image) having a high sharpness is to be reproduced, for example, the cubic spline interpolating operation may be used. In cases where a secondary image, which has a low sharpness and is smooth, is to be reproduced, for example, the B spline interpolating operation may be used.
In U.S. Pat. No. 5,048,105, the applicant proposed an interpolating operation method for an image signal, with which the sharpness of an interpolation image can be adjusted finely by, for example, weighting the corresponding coefficients of two interpolating functions, that provide different levels of sharpness, in accordance with a desired sharpness of the interpolation image, and adding the weighted coefficients to each other. With the proposed method, for example, in cases where the cubic spline interpolating operation function and the B spline interpolating operation function are employed as the two interpolating functions, that provide different levels of sharpness, the interpolation coefficients c.sub.k-1, c.sub.k, c.sub.k+1, and c.sub.k+2 for the cubic spline interpolating operation and the interpolation coefficients b.sub.k-1, b.sub.k, b.sub.k+1, and b.sub.k+2 for the B spline interpolating operation, which coefficients correspond to each other and are set for the respective original image signal components Y.sub.k-1, Y.sub.k, Y.sub.k+1, and Y.sub.k+2, are weighted and added to each other. By alteration of the weighting ratio (coefficient) .alpha., it is possible to obtain a secondary image having a desired level of sharpness, which is intermediate in the range from the highest sharpness to the lowest smooth sharpness.
Specifically, in cases where the interpolation coefficients for the cubic spline interpolating operation are represented by c.sub.k-1, c.sub.k, c.sub.k+1, and c.sub.k+2, and the interpolation coefficients for the B spline interpolating operation are represented by b.sub.k-1, b.sub.k, b.sub.k+1, and b.sub.k+2, weighted interpolation coefficients a.sub.k-1, a.sub.k, a.sub.k+1, and a.sub.k+2 are set as shown below. EQU a.sub.k-1 =(1-.alpha.)c.sub.k-1 +.alpha.b.sub.k-1 EQU a.sub.k =(1-.alpha.)c.sub.k +.alpha.b.sub.k EQU a.sub.k+1 =(1-.alpha.)c.sub.k+1 +.alpha.b.sub.k+1 EQU a.sub.k+2 =(1-.alpha.)c.sub.k+2 +.alpha.b.sub.k+2
wherein 0.ltoreq..alpha..ltoreq.1. PA1 i) obtaining an original image signal, which represents an original image and is made up of a series of original image signal components Yij, PA1 ii) linearly combining interpolation coefficients Bij and Cij, which correspond to each other and are set for each of the original image signal components Yij, in two different interpolating functions f and g for obtaining two interpolation images having different levels of sharpness, which functions are represented by Formulas (1) and (2), the linear combination being carried out with Formula (3), a new interpolation coefficient Aij being obtained from the linear combination, and PA1 iii) carrying out an interpolating operation on the original image signal components Yij by using an interpolating function h having the new interpolation coefficient Aij, which function is represented by Formula (4), an interpolation image signal being obtained from the interpolating operation, the interpolation image signal being made up of a series of image signal components, which occur at intervals different from those of the original image signal components Yij, EQU f=.SIGMA.Bij.multidot.Yij (1) EQU g=.SIGMA.Cij.multidot.Yij (2) EQU Aij=(1-.alpha.)Bij+.alpha.Cij (3) EQU h=.SIGMA.Aij.multidot.Yij (4) PA1 wherein the coefficient .alpha. in Formula (3) is set to be one of real numbers including a range smaller than 0 and/or a range larger than 1. PA1 a) a plurality of first look-up tables are prepared, each of the first look-up tables setting the frequencies and the corresponding responses R.sub.1 with respect to one of the two interpolating functions and for each of a plurality of different image size enlargement scale factors, PA1 b) a plurality of second look-up tables are prepared, each of the second look-up tables setting the frequencies and the corresponding responses R.sub.2 with respect to the other interpolating function and for each of a plurality of different image size enlargement scale factors, PA1 c) a response R.sub.1 of the one interpolating function with respect to an image size enlargement scale factor, which is desired for an interpolation image, and a response R.sub.2 of the other interpolating function with respect to the image size enlargement scale factor, which is desired for the interpolation image, are calculated by making reference to a first look-up table and a second look-up table, which correspond to the image size enlargement scale factor desired for the interpolation image, and PA1 d) the coefficient .alpha. is calculated from an operation carried out with Formula (5): EQU .alpha.=(R-R.sub.1)/(R.sub.2 -R.sub.1) (5) PA1 i) an original image signal, which represents an original image and is made up of a series of original image signal components Yij, is obtained, PA1 ii) interpolation coefficients Bij and Cij, which correspond to each other and are set for each of the original image signal components Yij, in two different interpolating functions f and g for obtaining two interpolation images having different levels of sharpness, which functions are represented by Formulas (1) and (2), are linearly combined with each other, the linear combination being carried out with Formula (3), a new interpolation coefficient Aij being obtained from the linear combination, and PA1 iii) an interpolating operation is carried out on the original image signal components Yij by using an interpolating function h having the new interpolation coefficient Aij, which function is represented by Formula (4), an interpolation image signal being obtained from the interpolating operation, the interpolation image signal being made up of a series of image signal components, which occur at intervals different from those of the original image signal components Yij, EQU f=.SIGMA.Bij.multidot.Yij (1) EQU g=.SIGMA.Cij.multidot.Yij (2) EQU Aij=(1-.alpha.)Bij+.alpha.Cij (3) EQU h=.SIGMA.Aij.multidot.Yij (4) PA1 a) a response input means for inputting a response R desired for the interpolation image, PA1 b) a plurality of first look-up tables, each of the first look-up tables setting the frequencies and the corresponding responses R.sub.1 with respect to one of the two interpolating functions and for each of a plurality of different image size enlargement scale factors, PA1 c) a plurality of second look-up tables, each of the second look-up tables setting the frequencies and the corresponding responses R.sub.2 with respect to the other interpolating function and for each of a plurality of different image size enlargement scale factors, and PA1 d) a coefficient calculating means for calculating a response R.sub.1 of the one interpolating function with respect to an image size enlargement scale factor, which is desired for an interpolation image, and a response R.sub.2 of the other interpolating function with respect to the image size enlargement scale factor, which is desired for the interpolation image, by making reference to a first look-up table and a second look-up table, which correspond to the image size enlargement scale factor desired for the interpolation image, and PA1 for calculating the coefficient .alpha. from an operation carried out with Formula (5): EQU .alpha.=(R-R.sub.1)/(R.sub.2 -R.sub.1) (5) PA1 in which Y.sub.k-1, Y.sub.k, Y.sub.k+1, and Y.sub.k+2 represent the original image signal components representing picture elements X.sub.k-1, X.sub.k, X.sub.k+1, and X.sub.k+2 in the original image, Y.sub.p represents the interpolated image signal component corresponding to an interpolation point X.sub.p located between the picture elements X.sub.k and X.sub.k+1, and a.sub.k-1, a.sub.k, a.sub.k+1, and a.sub.k+2 represent the interpolation coefficients, PA1 the method comprising the steps of: PA1 in which t, where 0.ltoreq.t&lt;1, represents the position of the interpolation point X.sub.p, the position being taken with respect to the picture element X.sub.k, that serves as a reference position, and in the direction heading toward the picture element X.sub.k+1, the lattice interval of the original image signal components (i.e., the picture element interval) being set to be equal to 1. PA1 in which Y.sub.k-1, Y.sub.k, Y.sub.k+1, and Y.sub.k+2 represent the original image signal components representing picture elements X.sub.k-1, X.sub.k, X.sub.k+1, and X.sub.k+2 in the original image, Y.sub.p represents the interpolated image signal component corresponding to an interpolation point X.sub.p located between the picture elements X.sub.k and X.sub.k+1,and a.sub.k-1, a.sub.k, a.sub.k+1, and a.sub.k+2 represent the interpolation coefficients, PA1 the method comprising the steps of: PA1 in which t, where 0.ltoreq.t&lt;1, represents the position of the interpolation point X.sub.p, the position being taken with respect to the picture element X.sub.k, that serves as a reference position, and in the direction heading toward the picture element X.sub.k+1, the lattice interval of the original image signal components (i.e., the picture element interval) being set to be equal to 1. PA1 in which Y.sub.k-1, Y.sub.k, Y.sub.k+1, and Y.sub.k+2 represent the original image signal components representing picture elements X.sub.k-1, X.sub.k, X.sub.k+1, and X.sub.k+2 in the original image, Y.sub.p represents the interpolated image signal component corresponding to an interpolation point X.sub.p located between the picture elements X.sub.k and X.sub.k+1, and a.sub.k-1, a.sub.k, a.sub.k+1, and a.sub.k+2 represent the interpolation coefficients, PA1 the method comprising the steps of: PA1 in which t, where 0.ltoreq.t&lt;1, represents the position of the interpolation point X.sub.p, the position being taken with respect to the picture element X.sub.k, that serves as a reference position, and in the direction heading toward the picture element X.sub.k+1, the lattice interval of the original image signal components (i.e., the picture element interval) being set to be equal to 1. PA1 in which Y.sub.k-1, Y.sub.k, Y.sub.k+1, and Y.sub.k+2 represent the original image signal components representing picture elements X.sub.k-1, X.sub.k, X.sub.k+1, and X.sub.k+2 in the original image, Y.sub.p represents the interpolated image signal component corresponding to an interpolation point X.sub.p located between the picture elements X.sub.k and X.sub.k+1, and a.sub.k-1, a.sub.k, a.sub.k+1, and a.sub.k+2 represent the interpolation coefficients, PA1 the apparatus comprising: PA1 in which Y.sub.k-1, Y.sub.k, Y.sub.k+1, and Y.sub.k+2 represent the original image signal components representing picture elements X.sub.k-1, X.sub.k, X.sub.k+1, and X.sub.k+2 in the original image, Y.sub.p represents the interpolated image signal component corresponding to an interpolation point X.sub.p located between the picture elements X.sub.k and X.sub.k+1, and a.sub.k-1, a.sub.k, a.sub.k+1, and a.sub.k+2 represent the interpolation coefficients, PA1 the apparatus comprising: PA1 in which Y.sub.k-1, Y.sub.k, Y.sub.k+1, and Y.sub.k+2 represent the original image signal components representing picture elements X.sub.k-1, X.sub.k, X.sub.k+1, and X.sub.k+2 in the original image, Y.sub.p represents the interpolated image signal component corresponding to an interpolation point X.sub.p located between the picture elements X.sub.k and X.sub.k+1, and a.sub.k-1, a.sub.k, a.sub.k+1, and a.sub.k+2 represent the interpolation coefficients, PA1 the apparatus comprising: PA1 in which Yij represents the original image signal components, and Bij and Cij represent the interpolation coefficients. In such cases, the term "linearly combining interpolation coefficients, which correspond to each other and are set for each of original image signal components" as used herein means that the operation is carried out with, for example, the formula EQU A12=(1-.alpha.)B12+.alpha.C12 PA1 wherein .alpha. represents all real numbers, on the interpolation coefficient B12 in the interpolating function f and the interpolation coefficient C12 in the interpolating function g which coefficients are for the original image signal component Y12 (in cases where i=1, and j=2).
In accordance with the thus obtained new interpolation coefficients a.sub.k-1, a.sub.k, a.sub.k+1, and a.sub.k+2, an interpolated image signal component Y.sub.p is calculated with Formula (38). EQU Y.sub.p =a.sub.k-1 Y.sub.k-1 +a.sub.k Y.sub.k +a.sub.k+1 Y.sub.k+1 +a.sub.k+2 Y.sub.k+2 ( 38)
An actual image is composed of the picture elements arrayed in two-dimensional directions. Therefore, the interpolation coefficient a.sub.k is represented as the interpolation coefficient Bij or Cij corresponding to each of two different directions (an i direction and a j direction) of an array of picture elements in the image.
With respect to the sharpness of the interpolation image, a wider variety of sharpness levels are often desired. For example, it is often desired to obtain an interpolation image having a sharpness higher than the sharpness of the interpolation image, which is obtained from the cubic spline interpolating operation alone. Also, it is often desired to obtain an interpolation image having a sharpness smoother than the sharpness of the interpolation image, which is obtained from the B spline interpolating operation alone.
However, with the interpolating operation method for an image signal, which is disclosed in U.S. Pat. No. 5,048,105, in cases where the cubic spline interpolating operation and the B spline interpolating operation are employed, the adjustment of sharpness can be carried out only within the range of the sharpness, which corresponds to the sharpest image obtained with the cubic spline interpolating operation, to the sharpness, which corresponds to the smoothest image obtained with the B spline interpolating operation. Therefore, the disclosed method cannot satisfy the demand for a wide variety of sharpness levels.